Charge and heat pumping in the Rice-Mele chain at finite temperature

A. A. Aligia (Centro At\'omico Bariloche; Argentina); Bariloche; CONICET; GAIDI; P. Roura-Bas

REVIEW 3 major objections 4 minor 44 references

Charge and heat pumping in the Rice-Mele chain at finite temperature

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For the non-interacting Rice-Mele chain, heat transported in a full pump cycle is not quantized, and for the common symmetric pump circuits it is exactly zero even where charge pumping remains quantized.

desk verdict Charge-pumping formulas are fine, but the zero-heat result is an artifact of a missing kinetic term in the energy current. read the letter →

arxiv 2411.15863 v1 pith:7OMDD5DV submitted 2024-11-24 cond-mat.mes-hall cond-mat.other

P. Roura-Bas , A. A. Aligia (Centro Atómico Bariloche , GAIDI , CONICET , Bariloche , Argentina) This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.other

keywords Rice-MelechainchargepumpingheattransportBerrycurvatureZakphasefinitetemperatureadiabatictopologicalpump

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper works out the charge and heat carried across a link of the non-interacting Rice-Mele chain during an adiabatic pumping trajectory at arbitrary filling and temperature. The transported quantities are expressed as double integrals of analytic functions built from the Berry curvature and the Fermi occupation of the instantaneous bands. At zero temperature and half filling, charge transport remains quantized and topological, but heat transport is not: it is weighted by the instantaneous on-site energy inside the integral, and for the symmetric elliptical circuits commonly used in the literature the heat transported per full cycle is identically zero. At infinite temperature or for completely filled bands, all transported charge and heat vanish.

What carries the argument

The central object is the heat current operator for the non-interacting chain, $J_Q=-\Delta\,J_p$, obtained from the energy continuity equation, with $J_p$ the particle current operator across a link between unit cells (Eq. (5)). The argument is carried by writing the expectation value of $J_p$ as the Berry curvature $\Omega_n(k,t)$ of the instantaneous Bloch bands (Eqs. (9)-(12)), so that the transported heat becomes the time integral of $\Delta(t)$ times the rate of change of the Zak phase $\gamma(t)$. The presence of $\Delta(t)$ inside the time integral is what breaks the topological quantization of heat, and the oddness of $\Delta(t)$ under $t\to T-t$ for symmetric circuits is what makes the integrated heat vanish over a full cycle.

What would settle it

Measure the energy transferred between two neighboring unit cells of an ultracold-atom Rice-Mele chain over one full symmetric elliptical pump cycle with $\Delta_0=0$, $v_1=\Delta_1=1$, $v_0=-1$: the paper predicts exactly zero net heat despite one unit of charge pumped, whereas the competing polarization-based picture of Ref. [34] predicts a nonzero, temperature-independent energy transfer; a direct calculation of the energy current that includes the explicit $\partial H_m/\partial t$ source term from the time-dependent parameters would also settle whether the vanishing-heat result survives once drive power is counted.

Watch

Extended reading notes

Core claim

The paper establishes that in the non-interacting Rice-Mele chain the heat transported through a link during an adiabatic process is, at half filling, $\Delta Q(t)=-(1/2\pi)\int_0^t dt'\,\Delta(t')\,\partial\gamma(t')/\partial t'$, where $\gamma$ is the Zak phase of the occupied band; the factor $\Delta(t')$ inside the time integral prevents the heat per cycle from being a topological integer. Consequently, the number of particles pumped in a closed cycle is still the integer winding number of the Berry phase, while the heat pumped is a $\Delta$-weighted winding that can vanish even when charge pumping is nontrivial. For any symmetric circuit with $\Delta_0=0$ in the standard elliptical parametrization, the time-reversal symmetry $t\to T-t$ of the integrand gives $\Delta Q(T)=0$ at all temperatures, whereas more general circuits transfer heat between the even and odd sublattices and the environment. The paper provides the analytic double-integral formulas, Eqs. (12), (17), and (19), for the charge, energy, and heat currents, and shows numerically that temperature smooths the sharp topological transitions in the transported charge into crossovers.

Load-bearing premise

The results assume that the heat current in the driven chain is exactly the particle current times the instantaneous on-site energy, as obtained from an energy continuity equation that does not include an explicit term for the power delivered by the external drive that changes the parameters in time.

Editorial extensions

If this is right

For the standard symmetric elliptical pump circuits with $\Delta_0=0$, the heat transported per full cycle is exactly zero at every temperature, even when the charge pumped is an integer.
For generic pump circuits with nonzero $\Delta_0$, heat is transferred between the even and odd sublattices of the chain and the environment, with a magnitude that decreases as temperature rises and vanishes at infinite temperature.
The heat pumped per cycle is not a topological invariant; only the zero-temperature, half-filled charge transport is quantized.
At infinite temperature or for completely filled bands, all transported charge, energy, and heat vanish for any adiabatic trajectory.
Finite temperature converts the sharp topological transitions in the pumped charge, seen as a circuit is displaced in parameter space, into smooth crossovers that are relevant for ultracold-atom experiments with unavoidable heating.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's claims, any circuit with $\Delta(t)$ odd under $t\to T-t$ should pump zero heat per cycle regardless of the trajectory shape, which would confirm the effect is a time-reversal symmetry property rather than a property of the chosen ellipse.
Beyond the paper's claims, these results imply that making a Thouless pump act as a heat engine requires breaking the time-reversal symmetry of $\Delta(t)$; symmetric modulation, the most common experimental choice, drives no heat.
A testable extension is to evaluate the interacting heat current with the time-evolving block decimation method the paper proposes; if the same symmetric circuit still gives zero net heat at finite interaction, the vanishing result would be robust beyond free fermions.
The discrepancy with Ref. [34] may stem from what is counted as heat: the continuity-equation current used here excludes the energy exchanged directly with the external drive, and that excluded part could appear as 'heat' in polarization-based definitions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Thanks for the report. I read the Rice-Mele paper carefully, and I think the reader's conditional verdict is close but actually too generous on the energy current. The missing piece is a real algebraic error, not a definitional preference.

What's genuinely here: the finite-temperature charge transport formula, Eq. (17), is a clean extension of the zero-temperature Berry curvature result. The contour plot in Fig. 1 is useful for reading off ΔN for arbitrary circuits, and the argument that charge transport loses its quantization at finite temperature is correct. The paper is well-organized, and the charge part should survive review.

The problem is Eq. (5). The energy current derived from dH_m/dt has two commutator contributions: the on-site part gives -Δ J_p, but the hopping part gives a second-neighbor term -i v w (c†_{2m-1}c_{2m+1} - h.c.). That term is missing. You can see it directly from the commutator of the link hopping with the hopping inside H_m. In k-space it contributes E_n Ω_n, not Δ Ω_n. For the symmetric circuits (Δ0=0), Δ(t) is odd under t→T-t while Ω_n is even, so Δ Ω integrates to zero by symmetry. But E_n is even, so E_n Ω does not vanish generically. That kills the headline zero-heat result. The ∂H_m/∂t term does integrate to zero over a cycle, so it cannot rescue the claim.

The reader flagged this as a weak assumption; I'd call it a load-bearing flaw. The abstract and Section V both rest on Eq. (19), which is only valid if you define the energy current without the kinetic part. That is not the standard continuity-equation definition of heat current, and the paper does not justify it.

Bottom line: the charge-pumping part deserves a serious referee; the heat-pumping part needs major revision, or at least an explicit, physically motivated definition that excludes kinetic energy transport. I would send it to peer review because the claim is important and the error is subtle enough that a careful referee would catch it. But I would not cite the heat result as it stands.

Referee Report

3 major / 4 minor

Summary. The manuscript studies adiabatic charge and heat pumping in the noninteracting Rice-Mele chain at finite temperature and arbitrary filling. The authors derive closed-form expressions for the transported charge and energy as double integrals over time and momentum of the Berry curvature weighted by Fermi functions. They find that at zero temperature and half filling the pumped charge is quantized for closed cycles, while the pumped heat is generally not quantized and depends on temperature; for symmetric elliptic circuits (Δ0=0) the heat transported per cycle vanishes identically. At infinite temperature or complete filling all pumped quantities vanish. The paper also discusses the difference between transport between unit cells and between sites within a unit cell.

Significance. If the central result is correct, it resolves a discrepancy with the earlier work of Hattori et al. and provides a practical criterion for when heat pumping accompanies topological charge pumping. The derivation is self-contained and uses standard adiabatic perturbation theory; the symmetry argument for the vanishing of heat for Δ0=0 is elegant and is supported by explicit circuit calculations. No free parameters are fitted, and the finite-temperature reduction of transport is explicitly demonstrated. The main weaknesses are technical gaps in the derivation of the heat current and in the handling of the chemical potential for arbitrary filling.

major comments (3)

[Section II, Eq. (5); Section III, Eq. (19)] The operator identity J_Eσ = [U n_{2m,barσ} − Δ] J_pσ is not the full energy current. A direct evaluation of dH_m/dt from the continuity equation also yields a kinetic (second-neighbor) term, e.g. i v w (c^†_{2m+2} c_{2m} − h.c.) for the link between cells m and m+1. In k-space this term is proportional to sin(k) times the identity in the two-band basis, so its expectation value integrates to zero over the Brillouin zone for the thermal distributions used (which are even in k). The paper does not state this, and as written Eq. (19) follows from an operator that is only the potential part of the energy current. The authors should either derive the full energy current and show that the kinetic contribution vanishes, or explicitly restrict their claim to the potential contribution. This is load-bearing because the vanishing-heat result for symmetric cycles is obtained from Eq. (19).
[Section III, Eq. (16)] The Fermi-Dirac distribution is written as f_n(k,t) = [exp(E_n(k,t)/k_B τ) + 1]^{-1}, which sets the chemical potential to zero. The paper claims to treat arbitrary filling, but for a general chemical potential the distribution must be [exp((E_n(k,t) − μ)/k_B τ) + 1]^{-1}. As printed, Eqs. (17) and (19) are valid only at half filling (μ = 0). This gap affects the claims in the abstract and in Section III about arbitrary filling and temperature; the authors should introduce μ explicitly and state which results are restricted to half filling.
[Section III, Eq. (12)] The expression for the Berry curvature is stated without derivation. Using the convention d = (v + w cos k, w sin k, Δ), the printed formula is consistent with the general two-band formula, and no sin(k) term is missing; the sin^2(k) contributions cancel after the dot product with d. The authors should include a brief derivation or at least state the convention for d_y, so that readers can verify the expression and the subsequent even/odd symmetry arguments.

minor comments (4)

[Abstract] The sentence 'We find that quantized transport is lost except in trivial cases' is misleading, because the paper shows that charge transport remains quantized at zero temperature and half filling for the usual circuits; the loss of quantization applies mainly to heat transport and to charge transport at finite temperature. Please rephrase to avoid confusion.
[Section V, Figs. 6 and 7] The statement that the results are only weakly dependent on the thermalization assumption is supported only for the energy transport in one circuit. A corresponding comparison for the finite-temperature charge transport would strengthen this claim, or the authors should state the range of parameters for which the insensitivity was checked.
[Section II] In the discussion of energy conservation, the text notes that the changes in energy on the two sides of the link are identical for U = 0; this point is correct but would benefit from a more explicit statement of the sign convention used for the currents, since J_p is defined as the rate of change of the left-region particle number.
[Section III, after Eq. (18)] The phrase 'provide and expression' contains a typo; it should read 'provide an expression'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the heat-transport formulas follow from a self-contained continuity-equation derivation and an explicit symmetry argument, with no fitted inputs.

full rationale

The derivation chain is self-contained. The energy current in Eq. (5) is obtained from the continuity equation for the left-half Hamiltonian H_m, and Eq. (19) follows by substituting the Berry-curvature expression for the particle current and the Fermi occupation factors. No parameter is fitted to a target heat or charge value. The statement that the transported heat vanishes for symmetric circuits with Δ0 = 0 is an explicit symmetry consequence: Δ(t) is odd and Ω(t) is even under t → T − t, so the integral in Eq. (19) vanishes over a full period. This is a derived result, not an input. Self-citations such as Refs. 15, 32, and 44 provide standard formulas or prior context, but they are not load-bearing for the central claim: the Berry-curvature expression (12) is obtained from the standard two-band formula, and the zero-temperature charge quantization is cited to the external textbook Ref. 11. The comparison with Hattori et al. (Ref. 34) is an external benchmark rather than a self-supporting citation. Any concern about whether Eq. (5) omits kinetic contributions to the energy current is a physical correctness issue, not a circularity, and no equation reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper has no free parameters fitted to data. The central claims rest on three modeling assumptions: (i) adiabatic evolution (Eq. 7), acknowledged at the end of Section VI; (ii) instantaneous thermalization, Eq. (16), with the slow-thermalization limit giving similar results; and (iii) the continuity-equation definition of the heat current, Eq. (5), which omits the explicit time-derivative term of the Hamiltonian parameters. The last assumption is the most consequential, since the vanishing-heat result for symmetric cycles is sensitive to it.

assumptions (4)

domain assumption Adiabatic theorem: the time-evolved state remains in the instantaneous eigenstate with first-order corrections (Eq. 7)
Justifies j(k,t)=Ω1(k,t); requires the drive frequency to be small compared with the instantaneous gap, as acknowledged at the end of Sec. VI.
domain assumption The occupancy of the bands is the instantaneous Fermi-Dirac distribution (fast thermalization) (Eq. 16)
Assumes the system rethermalizes at each instant at temperature τ; the paper also studies slow thermalization and finds similar results (Fig. 7).
domain assumption The energy (heat) current is given by the continuity-equation operator J_Eσ = (U n − Δ)J_pσ, Eq. (5), with no contribution from the explicit time dependence of the Hamiltonian parameters
The link current measures only particle-carried energy flux; the paper's heat-vanishing result for symmetric cycles is sensitive to this identification.
standard math Berry curvature formula (Eq. 11) from Ref 44 and the relation Ω2 = −Ω1 (Eq. 18) from particle-hole symmetry
Standard differential-geometry results for two-band models.

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Cite this review

Pith. "Pith review of Charge and heat pumping in the Rice-Mele chain at finite temperature." pith.science (2026). https://pith.science/paper/7OMDD5DV

@misc{pith2026241115863,
  author       = {Pith},
  title        = {Pith review of: Charge and heat pumping in the Rice-Mele chain at finite temperature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7OMDD5DV}},
  note         = {Machine review of arXiv:2411.15863}
}

read the original abstract

It is well known that quantized topological charge pumping takes place in the half filled Rice-Mele chain performing a closed cycle in parameter space. We extend previous studies to the case of charge and heat transport at arbitrary filling and temperature using the corresponding continuity equation with focus in the non-interacting case. The amount of charge and heat transported for any adiabatic time dependence of the parameters is given by a double integral of an analytical function. We find that quantized transport is lost except in trivial cases. In particular, for popular pumping circuits used which lead to quantized non-trivial charge transport at zero temperature, the heat transported in the cycle vanishes. For other pumping circuits, there is a heat transport among even and odd sites of the chain and the environment. As the temperature is increased, the transported charge and heat decrease and vanish at infinite temperature.

Figures

Figures reproduced from arXiv: 2411.15863 by the authors.

**Figure 2.** FIG. 2. Transported number of particles (top) and energy [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** FIG. 4. Total transported charge in the period ∆ [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 3.** Figure 3: FIG. 3. Same as Fig. 2 for ∆ [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]

**Figure 5.** Figure 5: FIG. 5. Transported charge between two unit cells (top) or [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Transported energy between two unit cells (top) or [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Same as Fig. 6 top for slow thermalization. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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